This seems counterintuitive. If the external pressure is equally applied around the egg, what forces the fluid to escape?
Looking at this 2020 video[0], it seems like the procedure might be:
1) Laser burns pinhole through shell.
2) Robotic pipette applies the marker chemical onto pinhole.
3) Pneumatic grabber picks up egg by its pinhole end (so the atmospheric pressure, relatively higher than the pressure inside the grabber, is pushing a little fluid out).
4) Egg is stored so marker chemical can be read later after it has undergone its reaction.
I found a 2018 video as well[1], but the machine looks a lot different and it's harder to tell how it works.
In the 2020 video, the shot that starts at 55 seconds shows the whole carousel. A pair of robot arms are using pneumatic grabbers to load eggs onto the carousel, they go under the high power laser, then pass under the pipette robot arms, and finally are removed from the carousel by a similar pair of robot arms with pneumatic grabbers (it is at this last step that I suspect the fluid passes through the shell).
It's more likely that the "applied pressure" is simply negative...